Let $\mu$ be a Lebesgue measure on the space $G$ of real symmetric $n \times n$ matrices (the Haar measure on the additive group of such matrices). For any $A \in G$ let $\chi_{A}(x)$ be its characteristic polynomial:

$$\chi_{A}(-x) = \det (A+xI) = x^n + a_{1}x^{n-1}+\ldots+a_{n-1}x + a_{n}.$$

Let $g(A) = ( a_1, \ldots, a_n )$ for any $A \in G$.

Consider a function $f \colon \mathbb{R}^n \to \mathbb{R}$ and define an integral

$$I =\int\limits_{H} f(g(A)) \, \mu(dA),$$

where $H$ is some subset of $G$ well characterized by $a_1,\dots,a_n$, for example the set of all positively defined matrices.

My question is how to reduce the integration with respect to $\mu$ to the integration with respect to the Lebesgue measure on the space of eigenvalues? I did't find an easy way. Maybe I have to use some generalized version of coarea formula to split the integration on the integration with respect to the Haar measure on $O(n)$ plus the integration on the space of eigenvalues?